# The Pentagon Problem

In this post I’ll talk about another favorite recreational math puzzle, the (in)famous “Pentagon Problem”.  First, though, I wanted to provide a solution to the Ghost Bugs problem from my last blog post.  The puzzle is the following:

You are given four lines in a plane in general position (no two parallel, no three intersecting in a common point). On each line a ghost bug crawls at some constant velocity (possibly different for each bug). Being ghosts, if two bugs happen to cross paths they just continue crawling through each other uninterrupted.  Suppose that five of the possible six meetings actually happen. Prove that the sixth does as well.

Here is the promised solution.  The idea (like in Einstein’s theory of general relativity) is to add an extra dimension corresponding to time.  We thus lift the problem out of the page and replace the four lines by the graph of the bugs’ positions as a function of time.  Since each bug travels at a constant speed, each of the four resulting graphs $g_i$ is a straight line.  By construction, two lines $g_i$ and $g_j$ intersect if and only if the corresponding bugs cross paths.

Suppose that every pair of bugs cross paths except possibly for bugs 3 and 4.  Then the lines $g_1, g_2, g_3$ each intersect one another (in distinct points) and therefore they lie in a common plane.  Since line $g_4$ intersects lines $g_1$ and $g_2$ in distinct points, it must lie in the same plane.  The line $g_4$ cannot be parallel to $g_3$, since their projections to the page (corresponding to forgetting the time dimension) intersect.  Thus $g_3$ and $g_4$ must intersect, which means that bugs 3 and 4 do indeed cross paths.

Cool, huh?  As I mentioned in my last post, I can still vividly remember how I felt in the AHA! moment when I discovered this solution more than 15 years ago.

# Riemann-Roch theory for graph orientations

In this post, I’d like to sketch some of the interesting results contained in my Ph.D. student Spencer Backman’s new paper “Riemann-Roch theory for graph orientations”.

First, a bit of background.  In a 2007 paper, Emeric Gioan introduced the cycle-cocycle reversal system on a (finite, connected, unoriented) graph G, which is a certain natural equivalence relation on the set of orientations of G.  Recall that an orientation of G is the choice of a direction for each edge.  A cycle flip on an orientation ${\mathcal O}$ consists of reversing all the edges in a directed cycle in ${\mathcal O}$.  Similarly, a cocycle flip consists of reversing all the edges in a directed cocycle in ${\mathcal O}$, where a directed cocycle (also called a directed cut) is the collection of all oriented edges connecting a subset of vertices of G to its complement.  The cycle-cocycle reversal system is the equivalence relation on the set of orientations of G generated by all cycle and cocycle flips.  In his paper, Gioan proves (via a deletion-contraction recursion) the surprising fact that the number of equivalence classes equals the number of spanning trees in G.  A bijective proof of this result was subsequently obtained by Bernardi. Continue reading

# Reduced divisors and Riemann-Roch for Graphs

In an earlier post, I described a graph-theoretic analogue of the Riemann-Roch theorem and some of its applications.  In this post, I’d like to discuss a proof of that theorem which is a bit more streamlined than the one which Norine and I gave in our original paper [BN].  Like our original proof, the one we’ll give here is based on the concept of reduced divisors. Continue reading

# Riemann-Roch for Graphs and Applications

I plan to write several posts related to the Riemann-Roch Theorem for Graphs, which was published several years ago in this paper written jointly with Serguei Norine.  In this post I want to explain the statement of the theorem, give some anecdotal background, and mention a few applications which have been discovered in recent years.

The Riemann-Roch Theorem

The (classical) Riemann-Roch Theorem is a very useful result about analytic functions on compact one-dimensional complex manifolds (also known as Riemann surfaces).  Given a set of constraints on the orders of zeros and poles, the Riemann-Roch Theorem computes the dimension of the space of analytic functions satisfying those constraints.  More precisely, if $D$ denotes the set of constraints and $r(D)$ is the dimension of the space of analytic functions satisfying those constraints, then the Riemann-Roch theorem asserts that

$r(D) - r(K-D) = {\rm deg}(D) + 1 - g$

where $g$ is the genus (“number of holes”) of the Riemann surface $X$, ${\rm deg}(D)$ is the total number of constraints, and $K$ is the “canonical divisor” on $X$.  See the Wikipedia page for much more information.